That's what I was thinking. You could get almost all of this pretty directly in Javascript by putting a callback function in an AsyncLocalStorage instance or, in other languages, in a thread local variable.
I was thinking this through more and realized that a callback directly in a thread local variable isn't quite enough: You need to keep a stack in the thread local variable. Every try-handle block pushes a callback in and pops it when exited. (Javascript's AsyncLocalStorage is naturally nestable so you don't need to manage a stack yourself with it.)